I am the first in the world

[alfa1]

http://news.yahoo.com/​spy-chief-nominated-romania​s-premier-182955470.html

Spy chief nominated as Romania's new premier

 

[Farsight]

This is still here?

Yep. RC is getting it, sol is wobbling, ben's got nothing, and Clinger has shown himself to be an ineffectual windbag who pretends he's dealt with the argument by posing homework questions.

"Get away with a sleight of hand." How ironic indeed!

I didn't cut anything out, my sleight-of-hand retort was in response to your ad-hominem suggestion that I'd tried to hide something.

It's as if I didn't hilite the most relevant part of post #51 just to make sure you couldn't possibly miss it...

I didn't miss it. What we're talking about here is a frozen star. It isn't a singularity at all. There are no infinities, it's merely a c=0.

That quote by Einstein in that post shows quite clearly what he thought about the Schwarzschild horizon: that spacetime does not simply end at the horizon and that the vanishing of coefficient of dt at r=2m (ρ=0) does not imply a genuine, physical singularity. Both of these in direct contradiction of your claims.

It's as much a singularity as a sphere of ice is a singularity to a fish frozen into it.

Though that might have been fine if it wasn't you who have continually insisted on Einstein's opinions on the matter, painted himself as being "with" Einstein, and his opponents as cranks for going "against" him.

Sol deserved the little lecture. Move on.

Fine, I'll repeat it again:

As above.

You are making a huge conceptual mistake. I suppose it's too much to ask for you to listen to W.D.Clinger's clear exposition on coordinates,

I replied to it in detail, see post #88, it was simple stuff concerning maps.

so I'll simply say this: if gravity was described by a scalar field, you might have been right. But gravity is a rank-2 tensor field: what's important is not just where you are, but how you are moving. This should have been clear from the metric, but it is especially explicit in the geodesic equation:
[latex]$\frac{d^2x^\alpha}{d\tau^2} = -\Gamma^{\alpha}_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}$[/latex]
giving the acceleration of a test particle in a gravitational field. This is not some modern invention. It's right there from the very genesis of GTR, in Einstein's first paper on the topic, and the Christoffel symbols themselves being called by Einstein the components of the gravitational field. Once one realizes that the effect of gravity depends not only on where the particle is but also how it is moving, it becomes clear that arguments like yours miss the mark completely.

That's got nothing to do with it. A bullet drops the same amount in a second as a dropped ball.

It's like discussing the behavior of electric charges but forgetting that magnetic fields exist (which give EM velocity dependence as well).

Electromagnetic fields exist. Remember what Minkowski said: "Then in the description of the field produced by the electron we see that the separation of the field into electric and magnetic force is a relative one with regard to the underlying time axis...". It's one field and two forces. Sounds to be as if we need to have a separate discussion about electromagnetism. One for another day.

I've outlined a simple proof from the geodesic equation that radial freefall matches Newtonian behavior in Schwarzschild r and proper time τ in post #64. More recently, ben m posted the cycloidal relations for this problem, which are again well-known for Newtonian gravitation (but obviously with ordinary radius and universal time for Newtonian case, rather than Schwarzschild r and proper time).

Your simple proof was wrong. When 2M/r = 1 at r = 2M you've got a division by zero in [latex]\[ \frac{dt}{d\tau} = \frac{e}{1-2M/r} \][/latex]. That's an undefined result, an infinity. But you just sail on past it. The finite proper time demands an infinite coordinate time, so it hasn't happened yet, and it never ever will.

I've derived the above using nothing but what is found in Einstein's original paper plus knowledge of Schwarzschild metric.

And even a blind man can see the problem with it.

Uncle Albert says you are wrong in just about every possible way. If you've actually read and understood his paper, you wouldn't have these problems.

Ditto.

 

[Reality Check]

Yep. RC is getting it, sol is wobbling, ben's got nothing, and Clinger has shown himself to be an ineffectual windbag who pretends he's dealt with the argument by posing homework questions.

RC has always got it, sol has always been solid, ben got that you have nothing to support your position and Clinger has shown himself to be a competent mathematician.
Farsight is continuing to show his inability to learn physics or mathematics.
Farsight has shown that he cannot even do homework assignments !

 

[Farsight]

Yes, that looks right. I didn't remember where the crossover was (as you can see from that plot, for a very low orbit the net effect would be a slowing), but I checked another source and it is indeed the case that GPS clocks run fast.

Good stuff. I'll check the crossover point re a discussion on another thread.

No we don't. If "speed of light is zero" were a real property of that location, it would be seen as such by anyone at that location. It's not.

Come on man, think. Nobody can see anything when the speed of light is zero. And it is a real property of that location: you know this from those parallel-mirror light clocks I keep showing you. And because a photon doesn't slow down like a cannonball whilst moving vertically up out of a gravitational field. We don't see photons travelling at some slower speed because they've been emitted by a giant star. In similar vein a photon emitted vertically from the event horizon isn't going to lose all its speed and obligingly fall back down. It isn't going to curve back down either, because it was emitted vertically. The black hole is black and the photon doesn't get out because its initial speed is zero.

Edited to add:

What Farsight isn't telling you, and appears not even to know, is that relativity has no well-defined concept of "yet". Your concept of "yet", like your concept of time itself, is observer-dependent; it depends on the chart (aka coordinate system/frame/patch) you choose to use. That's why the theory of relativity is called the theory of relativity.

Good catch. Farsight's saying "nothing has fallen into the black hole yet", and pretending it's an absolute statement, is basically the same mistake as saying "the back of the ladder hasn't entered the barn yet" in the famous thought-experiment.

It's no catch at all, merely cheap wordplay and obfuscation because the guy has no counter-argument. And if you want me to talk about the pole-in-the-barn paradox, it'll be my pleasure.

 

[dafydd]

http://news.yahoo.com/​spy-chief-nominated-romania​s-premier-182955470.html

Spy chief nominated as Romania's new premier

Wonderful news! Way to go for the spooks!

 

[Farsight]

Suppose someone falls into a black hole, holding a "light clock" (an arrangement of two mirrors held a fixed distance apart by a rigid piece of metal, with a light pulse bouncing back and forth between the mirrors).

This is going to be fun.

Q1: Assuming our victim is inside a sealed container and cannot see out, will she notice anything when she crosses the horizon? Specifically, measured with respect to the time scale of her own biological processes (or any other clock or process inside the container), will the time it takes for the light in the light clock to bounce back and forth change?

The stock answer is no, but this misses the crucial point: she hasn't crossed the event horizon yet. Even if she fell into the black hole a billion years ago, she hasn't crossed the event horizon yet. As we speak she's at the event horizon, frozen along with her light pulse, and she isn't noticing anything. She is in the same situation as a person who has been accelerated to the speed of light, her biological processes have stopped. We all know we can't actually accelerate our victim to the speed of light, I say it's because we're made of the darn stuff whilst you doubtless say it's because it would take infinite energy, but set that aside. Just go with the flow and assume we can. The person travelling at c is subject to infinite time dilation. For light to enter her eyes orthogonal to the direction of travel, light would have to be travelling faster than light. It can't happen. Light can't move in her frame, and nor can electromagnetic or biological processes. So it's not a case of she doesn't notice anything unusual, it's a case of she doesn't see or think.

A1: Nope - not if the black hole is large enough that tidal forces are small at the horizon, at least.

See above.

Q2: After having passed through the horizon, our intrepid investigator opens a porthole.

Whoa. Let's have your observer open a porthole as she falls into the black hole. She's looking back at the universe. What does she see just as she reaches the event horizon?

Can she still see stars etc. that are outside the hole? Can she receive signals sent by her friend in a spaceship hovering above the horizon?

A2: Yes to both. They might be distorted, but they are still visible. That would be quite hard to understand if time really stopped at the horizon...

You still don't get this sol, and it's important: time doesn't move. Things move. Things like light. Can you see time moving? No. Can you see light moving? Yes. It isn't time that stops at the event horizon. It's light. And remember electron spin and diffraction, and annihilation. Everything stops.

The truly fascinating this is how this is compatible with the description of the horizon as a hot membrane that I just gave above. Exercise for the reader: figure out how and why that's possible.

Where above? It isn't hot at all. Heat is an emergent property of motion. Motion is time-dilated out of existence. And I'm off to bed.

 

[ben m]

Come on man, think. Nobody can see anything when the speed of light is zero.

The speed of light isn't actually zero. There is a particular frame in which a particular ratio of coordinates equates to zero. That ratio is observer-dependent, and except in special contexts it doesn't actually represent the speed of light.

And it is a real property of that location: you know this from those parallel-mirror light clocks I keep showing you.

Nope. You keep analyzing that clock from the frame of special observers---one inertial observer in flat space, another maximally-accelerating observer near the horizon---and ignoring others. I have no idea why you think this is a good idea, or why you think that these observers get "the right answer" while my falling observer is somehow wrong.

Remember, Farsight, real GR can handle any observer, and can give consistent answers no matter what observer is chosen. The actual laws of GR include your rope-observer and his infinite blueshift, and your outside-observer and his nothing-has-fallen-in-yet redshift ... and it does so without contradicting, or ignoring, or devalueing, the experiences of my falling-straight-in observer ... the one who, according to a calculation you have failed to criticize or notice or debunk in any way, goes through the horizon in a finite proper time and can do local speed-of-light measurements all the way down.

 

[ben m]

The stock answer is no, but this misses the crucial point: she hasn't crossed the event horizon yet.

Not "yet" according to a clock at rest. She crosses the horizon in a finite time according to her own clock (and many other possible moving clocks).

Dammit, Farsight, this is an explicit solution of GR. It's in Misner Thorne and Wheeler with all of the coordinates, algebra, etc. laid out. I gave you the exact equation of motion---what do you think I did, made that up by throwing darts at a keyboard? Is there an error in the MTW derivation? Did they make a bad starting assumption?

Whoa. Let's have your observer open a porthole as she falls into the black hole. She's looking back at the universe. What does she see just as she reaches the event horizon?

She's moving very fast. Remember that collection of photons that your rope-observer thought was blueshifted like crazy? The infalling observer is nearly co-moving with those photons, and that motion "undoes" the blueshift. If she looks back, she sees an ordinary view of the Universe. Photons from behind enter her porthole with nonzero (but unremarkable, nonsingular) redshifts or blueshifts and with no special history/time compression. If she looks forward, she can see objects a finite distance in front of her; such objects are necessarily co-falling with her, having entered the black hole before she did. In front of those objects, she sees a rapidly-shrinking-as-she-falls-towards-it event horizon that she's doomed to keep approaching. This "Schwarzchild surface" has all of the properties of an event horizon---she sees things redshift as they approach it---except that she's doomed to continue approaching it.

You still don't get this sol, and it's important: time doesn't move. Things move. Things like light. Can you see time moving? No. Can you see light moving?

Nope, you're wrong. An observer falling through the horizon has an ordinary experience of space, time, motion, and light. Neither light nor time stop at the event horizon. You're wrong to think it does. You were tricked into thinking this by over-relying on a particular set of clocks.

ETA: http://casa.colorado.edu/~ajsh/singularity.html

 

[ben m]

The stock answer is no, but this misses the crucial point: she hasn't crossed the event horizon yet.

Farsight is using strictly circular reasoning.

F: "Everything stops at the event horizon."
S: "Here is an explicit GR solution showing something not-stopping at the event horizon."
F: "That solution can't be true, because everything stops at the event horizon."

 

[sol invictus]

Q1: Assuming our victim is inside a sealed container and cannot see out, will she notice anything when she crosses the horizon? Specifically, measured with respect to the time scale of her own biological processes (or any other clock or process inside the container), will the time it takes for the light in the light clock to bounce back and forth change?

The stock answer is no

So you agree that she experiences nothing at all unusual. She's sitting there looking at her light clock, and it's behaving exactly as it always did, she feels no forces, none of the other instruments inside her sealed lab read anything. For her, the moment when she arrives at the horizon (or let's say when she arrives at 0.0000000000000000000000000000000000001 mm outside it) is just a perfectly normal moment, after a finite number of light-clock-bounces.

But according to you we have to cut time off right at that perfectly ordinary, unremarkable moment. You can't ask what comes after that moment. You can't ask what bizarre law of physics could stop time right then, with nothing unusual happening, in a vacuum region of space.

You can't ask any of those questions, because you won't let yourself. You refuse to use any system of coordinates other than the Schwarzschild coordinates. And that (arbitrary and unjustified) choice prevents you from asking or answering any such question, because those coordinates only cover part of the spacetime.

 

[Farsight]

So you agree that she experiences nothing at all unusual.

No. She experiences what she'd experience if she was moving at c. She experiences nothing.

She's sitting there looking at her light clock, and it's behaving exactly as it always did, she feels no forces, none of the other instruments inside her sealed lab read anything. For her, the moment when she arrives at the horizon...

That moment hasn't come yet. Let's you and me wait a billion years, and that moment still hasn't come yet. It never comes. She's sitting there in front of her light clock, but light isn't moving in the clock, it isn't moving towards her eyes, and electromagnetic processes aren't moving in her instruments and in her brain.

(or let's say when she arrives at 0.0000000000000000000000000000000000001 mm outside it) is just a perfectly normal moment, after a finite number of light-clock-bounces.

Those finite number of light bounces takes an infinite coordinate time to occur. They never happen.

But according to you we have to cut time off right at that perfectly ordinary, unremarkable moment. You can't ask what comes after that moment.

Not so. The moment never comes. The supposed time after that moment is time after the end of time. Come on sol, look at the lefthand chart on post 135. The chopped-off top goes up to infinity. There's no getting past it.

You can't ask what bizarre law of physics could stop time right then, with nothing unusual happening, in a vacuum region of space.

Stop talking about stopping time. make the leap to stopping light, and then there's nothing bizarre about it at all. Think back to the optical clocks losing synchronisation at different elevations, simplified to parallel-mirror light clocks.

|------------------|
|------------------|

One light beam is going slower than the other, and c = √(1/ε₀μ₀). The vacuum impedance of space Z₀ = √(μ₀/ε₀) is increasing. Increase it so that it totally impedes the motion of light, and you stop light. Then nothing moves and clocks don't tick. Waves can't propagate through space any more. It's that simple.

You can't ask any of those questions, because you won't let yourself.

I don't fight shy of asking questions. I examined my own convictions long ago, and I work hard not to have any. Show me some scientific evidence that disproves something I say, and I will readily put my hands up, hang my head in shame, and say Sorry, I was wrong. I've done it before and I'll do it again. But that's what it takes, not some wan accusation concerning psychology.

You refuse to use any system of coordinates other than the Schwarzschild coordinates.

Not so. I challenge the use of neverneverland coordinates in cloud-cuckoo land. What we have here is an inverted version of Zeno's paradox: It takes an arrow one second to move half the distance between it and the target, and the arrow doesn't know it's slowing down. It never hits the target. The arrow's finite proper time is a fairytale.

And that (arbitrary and unjustified) choice prevents you from asking or answering any such question, because those coordinates only cover part of the spacetime.

Spacetime isn't something real, sol. Space is real, that black stuff between the stars. The thing that sustains waves and fields. Motion through it is real. Spacetime is a mathematical space in which there is no motion. You don't move through it. And when light doesn't move at the event horizon, all worldlines go straight up. Come on sol, we're nearly there with this. Examine your own psychology. You can't elucidate a clear explanation of why I'm wrong. Why not? Because I'm not. And I'm not some "my theory" guy pushing some damn fool hypothesis, this is just frozen-star stuff that you've never thought through. Think it through for yourself. Ask yourself whether the time really flows and whether the speed of light is really constant, and what gravitational time dilation really is. Don't stay locked in a position whereby you can't ask any of those questions, because you won't let yourself.

Och, I hope you find this thread stimulating.

 

[Farsight]

The speed of light isn't actually zero. There is a particular frame in which a particular ratio of coordinates equates to zero. That ratio is observer-dependent, and except in special contexts it doesn't actually represent the speed of light.

There are no frames ben. They don't exist. A reference frame isn't something you can point to up in the sky. Your observations and your measurements are affected by your motion, and by the thing we call gravitational potential, that's all.

Nope. You keep analyzing that clock from the frame of special observers---one inertial observer in flat space, another maximally-accelerating observer near the horizon---and ignoring others. I have no idea why you think this is a good idea, or why you think that these observers get "the right answer" while my falling observer is somehow wrong.

It's a good idea because frames aren't real whilst gravitational potential is. Light clocks tick slower when you're lower, and so do you. That's what actually happens, and light doesn't escape from a black hole. Now come on, address this:

And it is a real property of that location: you know this from those parallel-mirror light clocks I keep showing you. And because a photon doesn't slow down like a cannonball whilst moving vertically up out of a gravitational field. We don't see photons travelling at some slower speed because they've been emitted by a giant star. In similar vein a photon emitted vertically from the event horizon isn't going to lose all its speed and obligingly fall back down. It isn't going to curve back down either, because it was emitted vertically. The black hole is black and the photon doesn't get out because its initial speed is zero.

Remember, Farsight, real GR can handle any observer, and can give consistent answers no matter what observer is chosen.

The issue isn't whether it's real GR, ben, but whether the solution is real.

The actual laws of GR include your rope-observer and his infinite blueshift, and your outside-observer and his nothing-has-fallen-in-yet redshift ... and it does so without contradicting, or ignoring, or devalueing, the experiences of my falling-straight-in observer

Don't give me laws, give me scientific evidence and logic. It's the law just won't do.

...the one who, according to a calculation you have failed to criticize or notice or debunk in any way, goes through the horizon in a finite proper time and can do local speed-of-light measurements all the way down.

Point it out or repeat it and I'll knock it down like the rest of them.

Not "yet" according to a clock at rest. She crosses the horizon in a finite time according to her own clock (and many other possible moving clocks).

But in an infinite time according to my clock and your clock, and ET's clock. So she never crosses the horizon.

Dammit, Farsight, this is an explicit solution of GR. It's in Misner Thorne and Wheeler with all of the coordinates, algebra, etc. laid out.

And MTW is wrong. That's what this whole discussion is about. Your good book is wrong. Look at that lefthand chart and the chopped off infinity. A clock going infinitely slow along with an observer observing infinitely slow doesn't get you past the end of time. Hop skippity jump!

I gave you the exact equation of motion---what do you think I did, made that up by throwing darts at a keyboard? Is there an error in the MTW derivation? Did they make a bad starting assumption?

Yes!

Gotta go.

 

[ben m]

That moment hasn't come yet. Let's you and me wait a billion years, and that moment still hasn't come yet. It never comes.

Yes, there are coordinate systems in which she never gets there. The coordinate system where you and I are sitting still watching our clocks? That's one of them. Einstein's GR includes that.

She's sitting there in front of her light clock,

Hey, whoa! That's a different clock, in a different location, moving at a different speed. Why did you take results from your clock and decide that they would be true on this one? You forgot to do the transformation.

Those finite number of light bounces takes an infinite coordinate time to occur. They never happen.

... according to some clocks, and not others. Why don't you use GR to do this frame transformation? Is GR wrong?

The arrow's finite proper time is a fairytale.

Sorry, Farsight, Einstein appears to disagree with you. I have a spacetime that Einstein says is valid, a coordinate system in that metric that Einstein's math says is valid, and an equation of motion following Einstein's prescription. Which part is the "fairytale"?

When else does Einstein's math go so far wrong? How did, e.g., Hulse and Taylor predict the pulsar spindown rate? They used Einstein's math, straight out of the box, without doing a separate, nonmathematical "fairytale check" using the Farsight ask-a-distant-observer-and-don't-transform method?

 

[sol invictus]

No. She experiences what she'd experience if she was moving at c. She experiences nothing.

That moment hasn't come yet. Let's you and me wait a billion years, and that moment still hasn't come yet. It never comes. She's sitting there in front of her light clock, but light isn't moving in the clock, it isn't moving towards her eyes, and electromagnetic processes aren't moving in her instruments and in her brain.

Those finite number of light bounces takes an infinite coordinate time to occur. They never happen.

I didn't ask about the moment she crosses the horizon. I asked about the moment just before (say, 1 microsecond as measured by her clock), which does take place even in the coordinates you insist on.

You've agreed that our intrepid explorer will live up to that moment, and moreover will notice nothing whatsoever unusual. Not only will she not notice anything unusual, she won't be able to measure anything unusual, no matter how precise an experiment she does (again, apart from tidal forces - but even those don't necessarily tell her she's crossed a horizon). For her, up until that moment all clocks are ticking as usual, she's getting hungry at the usual rate, she can play tennis or a game of billiards as usual. She can even open a porthole and look outside at the stars behind her - for a large black hole she'll see nothing unusual, just some displacement of their location due to lensing.

So here we have this experimenter, living normally, noticing nothing - and yet, according to you, at some random instant that she cannot possibly determine by any experiment, time just..... stops. One can't ask about any moment afterwards. You even assert that there is no moment afterwards. According to you, time literally comes to an end, with no warning and no physical mechanism.

That's a totally absurd position on its face. (And let me point out again that it's also manifestly wrong, because any location in space is on the horizon in some set of coordinates.)

 

[Daylightstar]

...
that black stuff between the stars. The thing that sustains waves and fields. ...

Since you bring it up again, Farsight, how's that with gravitomagnetism and your "eleastic response" again?

Could you this time detail this "eleastic response", please?

 

[W.D.Clinger]

some homework problems for Farsight, epilogue 2

The homework problems I posted in Part 1 (see also this correction), Part 2, and Part 3 covered the basic definition of a spacetime manifold, Schwarzschild charts, and Lemaître charts. The relationship between Schwarzschild and Lemaître coordinates was first described in

Georges Lemaître. L'Univers en expansion. Annales de la Société scientifique de Bruxelles A 53, 51 (1933).​

I haven't been able to locate that paper online, and I don't read French anyway, but I found this English translation yesterday:

Georges Lemaître. The expanding universe. (Translated by M A H MacCallum.) General Relativity and Gravitation, volume 29, number 5, 1997, pages 641-680. DOI: 10.1023/A:1018855621348. Online at http://www.springerlink.com/content/k52l15p7g086mqu5/​

What we now know as Lemaître coordinates were introduced in section 11 of that paper. From its introduction:

In Section 11, we remove an apparent contradiction between Friedmann's theory and the solution of Schwarzschild's exterior problem. [...] We show that the singularity of the Schwarzschild exterior is an apparent singularity due to the fact that one has imposed a static solution and that it can be eliminated by a change of coordinates.

Section 11 begins by stating the Schwarzschild metric in Schwarzschild coordinates and referring to the singularity at r=2M. Then:

We intend to prove that the singularity of the field is not real and arises simply because one wanted to use coordinates for which the field is static.

In other words, the singularity is a mere coordinate singularity, akin to the coordinate singularities that occur at the poles of a polar spherical coordinate system. Lemaître then derives what we now call Lemaître coordinates and states the metric in those coordinates. Having proved the equivalent of my exercises 20 and 21, Lemaître concludes section 11 as follows:

The singularity of the Schwarzschild field is thus a fictitious singularity, analogous to that which appears at the horizon of the centre in the original form of the de Sitter universe.

Farsight's arguments are based upon his own personal misinterpretation of that "fictitious singularity" as a genuine singularity.

Farsight's like the guy who's got his atlas open to a chart that shows only the 48 contiguous states, and is using that chart to deny the existence of Hawaii and Alaska. When we tell him the next page contains a chart of all 50 states plus Mexico and Canada, he refuses to look.

I am not making this up:

When 2M/r = 1 at r = 2M you've got a division by zero in [latex]\[ \frac{dt}{d\tau} = \frac{e}{1-2M/r} \][/latex]. That's an undefined result, an infinity. But you just sail on past it. The finite proper time demands an infinite coordinate time, so it hasn't happened yet, and it never ever will.

Farsight hasn't done his homework. As shown in exercises 20 and 21, that division by zero occurs only when the metric is expressed in Schwarzschild coordinates. When exactly the same metric is expressed in Lemaître coordinates, there is no division by zero at the event horizon.

That moment hasn't come yet. Let's you and me wait a billion years, and that moment still hasn't come yet. It never comes. She's sitting there in front of her light clock, but light isn't moving in the clock, it isn't moving towards her eyes, and electromagnetic processes aren't moving in her instruments and in her brain.

Farsight hasn't done his homework, so he's using the wrong coordinate system here. As shown in exercise 23, Schwarzschild coordinates describe the situation as observed by an observer at infinity (or by an observer who's undergoing an outward acceleration that exactly counteracts the gravitational attraction of the black hole). As shown in exercise 22, Lemaître coordinates describe the situation as observed by the unfortunate woman who's free-falling into the black hole.

Not so. The moment never comes. The supposed time after that moment is time after the end of time. Come on sol, look at the lefthand chart on post 135. The chopped-off top goes up to infinity. There's no getting past it.

Farsight hasn't done his homework. As shown in exercises 20 and 21, the spacetime manifold extends beyond the open set on which Schwarzschild coordinates are defined. Had Farsight done his homework, he'd know Lemaître coordinates are finite and perfectly smooth at the event horizon, where Schwarzschild coordinates blow up because of a mere coordinate singularity.

Not so. I challenge the use of neverneverland coordinates in cloud-cuckoo land.

Farsight hasn't done his homework. Had he worked through the exercises, he'd know Lemaître coordinates describe exactly the same spacetime manifold whose outer portion is described by the Schwarzschild coordinates. He'd also know that Lemaître coordinates cover more of that manifold because they don't suffer from the coordinate singularity that limits Schwarzschild charts to the submanifold that lies outside the event horizon. Lemaître charts provide a perfectly smooth description that goes all the way down to the physical singularity at the heart of the black hole.

And MTW is wrong. That's what this whole discussion is about. Your good book is wrong. Look at that lefthand chart and the chopped off infinity. A clock going infinitely slow along with an observer observing infinitely slow doesn't get you past the end of time. Hop skippity jump!

Farsight hasn't done his homework. He flat-out refused to work through a simple set of exercises that would have taught him a few basic facts about general relativity and the spacetime manifold surrounding a black hole.

Because Farsight literally does not understand the most basic facts about spacetime manifolds, he is not qualified to judge whether MTW and other textbooks are right or wrong. He could learn something about relativity if he tried, but that might take a day or so of effort (or much longer if he has to learn calculus first). It's much easier for him to dismiss erudition out of hand, just because erudition and knowledge lead to conclusions that don't agree with his uninformed prejudices.

My exercises didn't do Farsight any good because he didn't do them. The exercises helped me, though, because I had never before used Lemaître coordinates.

Exercises 20 and 21 don't just tell you what's wrong with Farsight's argument: They show you. If you're proficient in high school algebra and differential calculus, you should be able to work through all of the exercises (except exercise 8) in a few hours. Those exercises won't make you into an expert on general relativity, but they'll give you a head start on Farsight.

 

[ben m]

W.D. Clinger, my hat's off to you; your three "homework" posts were very educational. To me, anyway. My MTW coursework is in the increasingly-distant past, and your posts remind me of things I *used* to be able to solve and prove.

As for Farsight, I think Upton Sinclair said the right thing. "It is difficult to get a man to understand something, when his salary ego depends upon his not understanding it."

 

[Farsight]

Yes, there are coordinate systems in which she never gets there. The coordinate system where you and I are sitting still watching our clocks? That's one of them. Einstein's GR includes that.

We sit here watching our clocks and looking through our telescopes, and we adopt the thing we call a coordinate system for our measurements. I've just been out for a cigarette, and I looked up at the night sky. I saw stars. They exist. The space between them exists too, and we're sure enough that black holes exist too. But coordinate systems don't. Here we are 13.7 billion years after what appears to be the beginning of the universe, at 18:22 GMT on 11th February 2012, and she hasn't got there yet. Let's call her Alice. There is no situation or coordinate system in which she has got there yet.

Hey, whoa! That's a different clock, in a different location, moving at a different speed. Why did you take results from your clock and decide that they would be true on this one? You forgot to do the transformation.

Whoa yourself. You said She crosses the horizon in a finite time according to her own clock, in post #156. I merely said She's sitting there in front of her light clock, so don't clutch at straws suggesting I forgot something. It doesn't matter what coordinate system you use, and regardless of transformations from one to another, you can't make Alice's collision event happen yet and not happen yet.

... according to some clocks, and not others. Why don't you use GR to do this frame transformation? Is GR wrong?

No, but some of the things people ascribe to GR are.

Sorry, Farsight, Einstein appears to disagree with you. I have a spacetime that Einstein says is valid, a coordinate system in that metric that Einstein's math says is valid, and an equation of motion following Einstein's prescription. Which part is the "fairytale"?

The point singularity in the middle of the black hole. It's always in the future.

When else does Einstein's math go so far wrong? How did, e.g., Hulse and Taylor predict the pulsar spindown rate? They used Einstein's math, straight out of the box, without doing a separate, nonmathematical "fairytale check" using the Farsight ask-a-distant-observer-and-don't-transform method?

It doesn't. The interpretation of the maths goes wrong when people sweep an infinity under the carpet. Now where's that expression of yours?

 

[ben m]

and regardless of transformations from one to another, you can't make Alice's collision event happen yet and not happen yet.

Sure I can. I've done it. It's an ordinary, boring disagreement-between-observers, of the sort that happens all the time in SR and GR. One observer thinks a quantity has diverged, another doesn't. As Sol has emphasized, you can find observers who have such disagreements even in flat spacetime.

 

[Farsight]

I didn't ask about the moment she crosses the horizon. I asked about the moment just before (say, 1 microsecond as measured by her clock), which does take place even in the coordinates you insist on.

OK.

You've agreed that our intrepid explorer will live up to that moment, and moreover will notice nothing whatsoever unusual. Not only will she not notice anything unusual, she won't be able to measure anything unusual, no matter how precise an experiment she does (again, apart from tidal forces - but even those don't necessarily tell her she's crossed a horizon). For her, up until that moment all clocks are ticking as usual, she's getting hungry at the usual rate, she can play tennis or a game of billiards as usual.

Agreed. But as she approaches the event horizon, she's subject to increasing gravitational time dilation. Her awareness is slowing down, and she doesn't notice it. She doesn't notice anything unusual as she stops noticing everything.

She can even open a porthole and look outside at the stars behind her - for a large black hole she'll see nothing unusual, just some displacement of their location due to lensing.

There's some issues with this, but let's come back to it another time.

So here we have this experimenter, living normally, noticing nothing - and yet, according to you, at some random instant that she cannot possibly determine by any experiment, time just..... stops. One can't ask about any moment afterwards. You even assert that there is no moment afterwards. According to you, time literally comes to an end, with no warning and no physical mechanism.

I told you that the motion of light ceases due to total vacuum impedance in post #159. Please don't try to suggest that I'm advancing wild hypotheses without scientific support. For your convenience I'll reiterate what I said:

Stop talking about stopping time. Make the leap to stopping light. There's nothing bizarre or mysterious. Think back to the optical clocks losing synchronisation at different elevations, simplified to parallel-mirror light clocks.

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One light beam is going slower than the other, and c = √(1/ε₀μ₀). The vacuum impedance of space Z₀ = √(μ₀/ε₀) is increasing. Increase it so that it totally impedes the motion of light, and you stop light. Then nothing moves and clocks don't tick. Waves can't propagate through space any more. It's that simple.

That's a totally absurd position on its face. (And let me point out again that it's also manifestly wrong, because any location in space is on the horizon in some set of coordinates.)

It isn't manifestly wrong. Those coordinate systems are abstract mathematical artefacts. Black holes exist, space exists, light exists, gravitational fields exist. All these things are manifest. Coordinate systems are not.